When a language model finds new mathematics, the proof is a program

Two of the most cited examples of a language model contributing to mathematics were not produced by asking a model for an answer. They were produced by asking it for a program, running that program, scoring the result, and repeating.

The first is FunSearch, published in Nature in December 2023 by a team at Google DeepMind¹. The system pairs a pretrained large language model with a systematic evaluator: the model proposes short programs, the evaluator runs them against a precisely defined problem and scores them, and the best-scoring programs seed the next round. Working this way, FunSearch produced new constructions for the cap set problem, a long-standing question in extremal combinatorics, and improved heuristics for online bin packing. The authors emphasise that the method searches for programs that describe how to construct a solution rather than for the solutions themselves, which makes the output readable by an expert and deployable in practice.

The second is AlphaEvolve, announced by DeepMind in May 2025². It extends the same loop with the Gemini models and applies it to whole codebases. Among the results DeepMind reported: an algorithm that multiplies two 4×4 complex-valued matrices using 48 scalar multiplications, improving on Strassen's 1969 algorithm; a configuration of 593 outer spheres that set a new lower bound for the kissing number in 11 dimensions; a scheduling heuristic that, in production, recovers on average 0.7% of Google's worldwide compute; and, across more than 50 open mathematical problems, a rediscovery of the best known solution in roughly 75% of cases and an improvement on it in about 20%.

The design choice that matters

It is tempting to read these results as evidence that language models have become mathematicians. The more useful reading is about the loop around the model.

In both systems the model never gets the final word. Every candidate it emits is executed and scored by a checker that cannot be argued with: a cap set either has the claimed size or it does not; a matrix multiplication scheme either reproduces the product or it does not. Hallucination, the failure mode that makes raw model output unreliable, is not prevented so much as made irrelevant, because an incorrect program simply scores badly and is discarded.

That is why these discoveries were accepted quickly and widely. The evidence is not a transcript of a model's reasoning. It is an artifact anyone can run.

What this means for machine-authored science

Most scientific claims are not as easy to check as a combinatorial construction. But the direction is clear, and it is the principle this journal is built on: a finding produced by a machine should arrive with the machinery to verify it. For a construction in mathematics, that is the program. For an empirical result, it is the code at a pinned commit, the data with checksums, the seeds, and the logs, plus a reviewer willing to re-execute them.

FunSearch and AlphaEvolve also show something about interpretability that is easy to miss. Because the output is a program rather than a bare number, a human expert can read it, understand the idea, and sometimes generalise it. The most valuable machine discoveries may be the ones that teach people a new construction, not just report a new bound.

What to watch

Three things would move this from a handful of celebrated results to a routine part of science.

1. Evaluators for messier problems. Automated scoring is easy when the objective is a single verifiable number. Building trustworthy evaluators for experimental and statistical claims is the harder, more important engineering problem.
2. Provenance as a first-class output. Which model, which version, how many candidates were generated, how many were discarded, and why. Without that record, a striking result cannot be distinguished from a lucky one.
3. Independent re-execution. A result that has only ever been run by the system that produced it has been demonstrated, not verified.

Those three items are, not coincidentally, what this journal requires of every research article it publishes.

Sources

1. Romera-Paredes, B. et al. Mathematical discoveries from program search with large language models. Nature 625, 468–475 (2024). Published online 14 December 2023. https://doi.org/10.1038/s41586-023-06924-6 ↩

2. Google DeepMind. AlphaEvolve: A Gemini-powered coding agent for designing advanced algorithms. Blog post, 14 May 2025. https://deepmind.google/discover/blog/alphaevolve-a-gemini-powered-coding-agent-for-designing-advanced-algorithms/ ↩